enhancement a

7/27/2019 Enhancement A

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Image Enhancement in the

Raul Queiroz Feitosa

Gilson A. O. P. Costa


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Enhancement in the S atial

DomainContents

Introduction

Background

Histogram Processing

oca n ancement

Spatial Filters Smoothing Filters

Sharpening Filters

9/4/2013 Enhancement in the Spatial Domain 2


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Introduction

Two Groups of Enhancement Techniques:

In the Spatial DomainBased on the direct manipulation of the pixels in an image.

In the Fre uenc Domain

Based on modifying the Fourier Transform of an image.

g a

imagedigital

image

Acquis ition Enhancement SegmentationFeature

extractionRecognition

Post-

processing



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Back round

Notation:

(x,y) =T[f(x,y) ]

where: y, ,

g(x,y) is the output image, and (x,y)

a neighborhood of (x,y).

x



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Back round

Two groups of approaches:

Point Processing: when the neighborhood is of size

11, it becomes agray level(orintensity ormapping)transformation function of the form:

s =T(r)

where rands denote the gray level atf(x,y) andg(x,y)at an oint x, .

Mask-Based: uses small arra s called mask whosecoefficients determine the nature of the process.



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Point Processin

Image Negatives

l-s

raylev

e

T(r)

Output

-



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Point Processin

Contrast Stretching

(r2,s2)

l-s

T(r)raylev

e

Output

1, 1

-



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Point Processin

Power Law Transformation (Gamma Correction)

Correo Gama

where

l-s

=0.04

=0.1

=0.2

raylev

e=0.4

=0.67

=1

Output =1.5

=2.5

=5.0

-

=10

=25



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Point Processin

Power Law Transformation (Gamma Correction)

=0.04 Correo Gama

where

=0.1

=0.2

=0.4l-s

=0.04

=0.1

=0.2

=0.67

=1raylev

e=0.4

=0.67

=1=1.5

=2.5

=5.0Output =1.5

=2.5

=5.0

=10

=25

-

=10

=25



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Point Processin

Brightness Adjustment

10tosubjected, = sbrs

l-s

raylev

e

Output


npu gray eve - r


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Histo ram Processin

Definition

The histo ram of a di ital ima e with ra levelsin the range [0,L-1] is a discrete function

where:

rkis the kth gray level,

nkis the number of pixels having gray level rk



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Histo ram Processin

Example: dark image



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Histo ram Processin

Example: bright image



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Histo ram Processin

Example: image with low contrast



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Histo ram Processin

Example: image with high contrast



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Histo ram Processin

When the pixel counts are normalized and pixel

intensity (gray value) is considered a random variable,histogram is analogous to a PDF

w ere

rk

is the kth gray level,

nk s t e num er o p xe s av ng gray eve rk, an

n is the number of pixels in the image.



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Histo ram Processin

Problem formulation: design a functions =T(r) that

has uniformly distributed gray levels.



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Histo ram Processin

Letpr(r) andp

s(s) be the histogram respectively of the

Assume that rands represent the continuous gray levels

, .

Search for the transformation function T(r) satisfying

a) T(r) is a single-valued and monotonically increasing in [0,1]

b) 0 T(r) 1 for 0 r1c) the inverse function T-1(s) must also meet the above conditions



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Histo ram Processin

From basic probability theory, ifpr(r) and T(r) are

ons er ng t e umu at ve str ut on as t e

transformation function:



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Histo ram Processin

From basic probability theory, ifpr(r) and T(r) are

ca ng up t e trans ormat on unct on to t e actua gray

levels we have:



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Histo ram Processin



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Histo ram Processin

In the continuous case

In the discrete case

fork= 0,1,,L-1 where n is the total number of pixels



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Histo ram Processin

rk nk p(rk) =nk/n

r0 = 0 790 0.191 .

r2 = 2 850 0.21

r3 = 3 656 0.16

r4 = 4 329 0.08

r5 = 5 245 0.06

r6 = 6 122 0.03

r7 = 7 81 0.02

= , n= x =



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Histo ram Processin

s0 = 1.33 1 s4 = 6.23 6

s1 = 3.08 3 s5 = 6.65 7s2 = . s6 = . s3 = 5.67 6 s7 = 7.00 7



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Histo ram Processin

A histogram is and approximation of a PDF and no new

histograms are rare.

equalization results in uniform histogram.

dynamic range.

.



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Histo ram Processin

r

Equalization



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Histo ram Processin

Problem formulation: design a transformation function

=

output image has a specified histogram.



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Histo ram Processin

Letpr(r) andps(s) be the histogram of the input image and the

.

s

referenceinput

H(r)

s

G-1(z)

r un orm

pr

(r) ps(s)

s = T(r) = G-1[H(r)]



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Histo ram Processin

Histogram Matching

image 1999

mage w e

histogram of image 2001

mage



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Local Enhancement

The parameters of most functions presented so far are

computed based on the whole image. To enhance details in small image areas (example):

1. Select a neighborhood, whose center moves from pixel to pixel

over e mage.

2. At each position find the transformation function based on the

.

3. Apply the function to the pixel at the center of the neighborhood.

4. Move the center to the next ixel and re eat ste s 2 and 3 till allimage is covered.



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Local Enhancement

Example: local histogram equalization

Original Image After Global Equalization After Local Equalization



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Re ion of Interest Processin

(ROI) Logical matrices called masks can be used to define regions

in an image (roi) where an operator is to be applied. Example: blurring uninteresting regions

input image mask (roi) output image



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Local Enhancement

Global statistics: the nth moment about the mean

where( ) ( ) ( )ii

n

in rpmrr

=

=0

( )

=

=0i

ii rprm



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Local Enhancement

Local statistics: let (x,y) be the pixel coordinates and Sxy a

nei hborhood with a iven size around x .

average gray level around Sxy= ttS rprm

contrast around S y= tStS rpmr22

( ) xySts,

xySts ),(



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Local Enhancement

Example: enhance SEM image (tungsten filament

Enhance only dark areas

,

but not constant/uniform areas



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Local Enhancement


Enhance only dark areas:

GS Mkm xy 0

MG is the global mean

xySm is the local mean

k0 is positive, less than one

Sxy is 3x3



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Local Enhancement


Enhance low contrast areas:

GS Dkxy 2

xyS

DG is the global standard dev

is the local standard dev

k2 is positive, less than one

Sxy is 3x3



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Local Enhancement


Enhance non uniform areas:

xySGDk 1

xyS

DG is the global standard dev

is the local standard dev

k1 is positive, less than k2Sxy is 3x3



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Local Enhancement



,




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Local Enhancement



,


E=4;k0=0.4;k1=0.02;k2=0.4}



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Smoothin Filters

Aplications:

Removal of small details prior to large object extraction.

Noise reduction. Side Effect

It blurs the image.

moo ng mas s

All elements are non-negative, and

sum u to 1.

Masks 1 11

1 111/9

1 17

7 7541/881 11 1 17



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Smoothin Filters

Examples of applying an averaging filter:

original 33 55



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Smoothin Filters

Order Statistic Filters

are non-linear s atial filters whose res onse isbased on ordering (ranking) the pixels contained

,

then replacing the value of the center pixel with.


S hi Fil


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Smoothin Filters

Order Statistic Filters:

Median Filterreplaces the value of a pixel by the median of the gray

levels in its nei hborhood

A median of a set of values is such that half of thevalues in the set is lower than and hal o the values

in the set is greater than or equal to .

Percentile Filterreplaces the value of a pixel by n-th percentile of the

ra levels in its nei hborhood.


S hi Fil


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Smoothin Filters

Original image Image with salt-and-pepper noise


Ad i Fil


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Ada tive Filter

The out ut x of a ada tive filter de ends on

the statistical characteristics of the input image

xycentered at (x,y).

variance of the noise (?)

[ ]Lmyxfyxfyxg = ),(),(),( 22

variance in Sxy mean in S


Ad i Fil


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Ada tive Filter

is zero: no noise, nothing should change2

is high relative to : no change, probably an edge

is similar to : reduce noise b avera in2

2

2

2

L

is less than : must avoid/treat negative output

L

2

2

L

variance of the noise (?)

[ ]Lmyxfyxfyxg = ),(),(),( 22

variance in Sxy mean in S


Ad ti Filt


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Ada tive Filter

Exam le

image corrupted by

noise of zero mean

and variance 1000

noise reduction filter

mean filter


Sh i Filt


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Shar enin Filters

A lications Highlight of fine details.

acquisition method.

Side Effect

Emphasizes the noise.


Sh r nin Filt r


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Shar enin Filters

Accomplished by spatial differentiation:

Approximation for the 1 derivative

( ) ( ) ( )( ) ( )xfxf

x

xfxxf

x

xf+

+=

1lim

( ) ( ) ( )[ ] ( ) ( )[ ]2 + xxfxfxfxxfxf

( ) ( ) ( )121

22

++

xfxfxf

xx ox


Shar enin Filters


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Shar enin Filters


Ed e Detection


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Ed e Detection-1 0 1

-1 0 1

-1 0 1 3

4

2

1

0

4

pixel values of

horizontal line

Prewitt mask

2

1

0

3

first derivative-1

-2

-3

-4

2

1

3

4

-1

-2

-3

0 second derivative

9/4/2013 Image Segmentation 53

-4

Shar enin Filters


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Shar enin Filters

From the previous slide we may conclude:

1. First-order derivatives generally produce thickeredges in the image,

2. Second-order derivatives have a stron er res onse at

the fine details,3. First-order derivatives enerall have a stron er

response to a gray-level step, and

4. Second-order derivatives have a double-res onse atstep changes in gray level.


Shar enin Filters


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Shar enin Filters

Laplacian Filter22

2 ff 22 yx

( ) ( ) ( )yxfyxfyxfxf

,1,2,12

2

++

( ) ( ) ( )1,,21,2

2

++

yxfyxfyxf

y

f

we o ta n

( )[ ( ) ( ) ( ) ( ) ]yxfyxfyxfyxfyxff ,41,1,,1,12 +++++=


Shar enin Filters


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Shar enin Filters

Laplacian Filter: example of masks

1 -4 1 4 -20 41 -8 1


Shar enin Filters


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Shar enin Filters

Laplacian based filter : an example

if center coefficient 0( ) ( )

+=

yxfCyxfyxfs

,,,,,

2

)(xf

e s arpen ng e ect n -

)(2 xf

)()( 2 xfxf s aper

transition


Shar enin Filters


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Shar enin Filters

Laplacian based filter: a mask example

0 00 1 00 -C 00

=1 000 00

-4 11

1 00

4C+1 - C- C

- C 00

-C


Shar enin Filters


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Shar enin Filters

Imagem Original C=4C=2

C=10C=6 C=8


Shar enin Filters


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Shar enin Filters

Unsharp masking

Subtracts a blurred version of an image from the imageitself

( ) ( ) ( )yxfyxfyxfum ,,, =

Example of a mask output of a smoothing filter

0 00

--

-1/9 -1/9-1/91 11

=

0 00 -1/9 -1/9-1/9

-

1 11


Shar enin Filters


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Shar enin Filters

Filter High-Boost

It is a generalization of unsharp masking, given by

( ) ( ) ( ) 1for,,, = AyxfyxAfyxfhb

output of a sharpening filter

,,, = shb

If we elect to use the Laplacian filter for sharpening

if center coefficient 0( )

( ) ( )

+=

yxfyxAf

yxyxyxfhb

,,

,,,

2


Shar enin Filters


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Shar enin Filters

Filter High-Boost: example

or g na

imagelaplacian

high-boost

with A=1

high-boost

with A=1.7


Shar enin Filters


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e e s

The Gradient

= xf

y

2/122

==

ff

x



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Shar enin Filters


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Gradient: example

original image |Gy| (sobel)|Gx| (sobel)


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